Approximate Solution of Volterra-Stieltjes Linear Integral Equations of the Second Kind with the Generalized Trapezoid Rule

Abstract

The numerical solution of linear Volterra-Stieltjes integral equations of the second kind by using the generalized trapezoid rule is established and investigated. Also, the conditions on estimation of the error are determined and proved. A selected example is solved employing the proposed method.

Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

“…Differential and integral equations theory considering fractional order are relevant in mathematics nowadays, which have numerous applications in various fields, physics, mechanics, control theory, engineering, electrochemistry, bioengineering, viscoelasticity, porous media [1] [2] [3]. Solution of the nonlinear integral equation of Volterra-Stieltjes type, and the method is based on an equivalence relation between the fractional differential equation, and Volterra-Stieltjes integral equation of the second kind was also reported in our previous works [4] [5].…”

Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

“…In Banas et al (2000) and Federson and Bianconi (2001), quadratic integral equations of Urysohn-Stieltjes type and their applications are investigated. Various numerical solution methods for integral equations are presented in Asanov et al (2011a, b), Asanov and Abdujabbarov (2011), Asanov et al (2016), Delves and Walsh (1974), Federson et al (2002). In particular, the generalized trapezoid rule and the generalized midpoint rule to evaluate the Stieltjes integral approximately by employing the notion of derivative of a function by means of a strictly increasing function (Asanov et al 2011a, b;Asanov 2001), the generalized trapezoid rule for linear Volterra-Stieltjes integral equations of the second kind (Asanov et al 2016), and the generalized midpoint rule for linear Fredholm-Stieltjes integral equations of the second kind (Asanov and Abdujabbarov 2011).…”